Conley Conjecture Revisited

TL;DR

This paper proves that certain symplectic manifolds with finitely many simple periodic orbits must have a specific spherical homology class, extending known results and introducing new techniques involving spectral invariants and Lusternik-Schnirelmann theory.

Contribution

It establishes a new criterion linking finite periodic orbits to spherical homology classes and develops a novel approach using spectral invariants and Lusternik-Schnirelmann theory.

Findings

Manifolds with finitely many simple periodic orbits have a spherical homology class of degree two.

Sequence of mean spectral invariants for iterations never stabilizes under certain conditions.

Action gaps between spectral invariants are bounded, implying generic existence of infinitely many simple periodic orbits.

Abstract

We show that whenever a closed symplectic manifold admits a Hamiltonian diffeomorphism with finitely many simple periodic orbits, the manifold has a spherical homology class of degree two with positive symplectic area and positive integral of the first Chern class. This theorem encompasses all known cases of the Conley conjecture (symplectic CY and negative monotone manifolds) and also some new ones (e.g., weakly exact symplectic manifolds with non-vanishing first Chern class). The proof hinges on a general Lusternik-Schnirelmann type result that, under some natural additional conditions, the sequence of mean spectral invariants for the iterations of a Hamiltonian diffeomorphism never stabilizes. We also show that for the iterations of a Hamiltonian diffeomorphism with finitely many periodic orbits the sequence of action gaps between the "largest" and the "smallest" spectral invariants remains bounded and, as consequence, establish some new cases of the $C^\infty$-generic existence of infinitely many simple periodic orbits.